Theorem dfon2lem1 36669

Description: Lemma for dfon2 36678. (Contributed by Scott Fenton, 28-Feb-2011.) (Proof shortened by SN, 5-Oct-2026.)

Assertion

Ref	Expression
dfon2lem1	⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

Proof of Theorem dfon2lem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		truni 5227	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)})
2		simp2 1155	. . . . 5 ⊢ ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) → Tr 𝑥)
3	2	ss2abi 3990	. . . 4 ⊢ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ⊆ {𝑥 ∣ Tr 𝑥}
4	3	sseli 3905	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → 𝑦 ∈ {𝑥 ∣ Tr 𝑥})
5		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
6		treq 5218	. . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
7	5, 6	elab 3632	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ Tr 𝑥} ↔ Tr 𝑦)
8	4, 7	sylib 221	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦)
9	1, 8	mprg 3082	1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

This proof depends on syntax axioms:   ∧ w3a 1103   ∈ wcel 2144  {cab 2738  ∪ cuni 4866  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3897  df-uni 4867  df-iun 4952  df-tr 5212

This theorem is used by:  dfon2lem3  36671  dfon2lem7  36675